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CALCULUS NOTES, WEEK PRECEDING SPRING BREAK
During the week, we continued discussing how we determine on what intervals a funciton is
increasing or decreasing, and how to find where it has a local maximum or a local minimum. We
also introduced analogous concepts: A function can be convex (sometimes called concave up) or
concave (sometimes called concave down). A point at which a function switches from convex to
concave, or from concave to convex, is called an inflection point. Given a function, we need to know
how to find on what intervals the function is convex and on what intervals it is concave, and where
the inflection points are.
0.1. Convexity and concavity. Assume the function f is differentiable. Then f is convex on an
interval if for any point in the interval, if we graph the function and the tangent line at that point
together, the line will lie below the graph of the function. In class, we argued why this happens
if and only if the derivative f 0 is increasing on this interval. Therefore we can find out where f is
convex by finding out where f 0 is increasing. But we remember from last week that we can find out
where a differentiable function is increasing by finding out where its derivative is non-negative. So
if f has two derivatives, we can find out where it is convex by finding out where f 0 is increasing,
which we can find by finding out where the second derivative f 00 is bigger than or equal to 0.
Similarly, we say that f is concave on an interval if for any point in the interval, if we graph
the function and the tangent line at that point together, the line will lie above the graph of the
function. To find out where this happens, we need only to look for the intervals where f 00 6 0.
The process for finding the intervals of convexity/concavity and for finding inflection points is
similar to finding the intervals of increasing/decreasing and for finding local maxima/minima. The
only difference is where increasing/decreasing and local maxima/minima uses the first derivative
f 0 , intervals of convexity/concavity and inflection points uses the second derivative f 00 .
The steps are as follows:
Given f , we find f 00 . We find all the solutions to f 00 (x) = 0, and we find all the points where
f 00 (x) does not exist. Let us list these points from left to right as c1 , c2 , . . . , cn . That is, the list
c1 , . . . , cn is a list (in increasing order) of the points where f 00 is either zero or does not exist.
If the domain of f is all real numbers, then we need to consider the intervals (−∞, c1 ), (c1 , c2 ),
. . ., (cn , ∞). If the domain of f is not all real numbers, then it will typically be an interval. If the
left endpoint of the domain is a and the right endpoint is b, we must consider (a, c1 ), (c1 , c2 ), . . .,
(cn , b) (that is, we start at a instead of −∞, and we end at b instead of ∞).
We then determine whether f 00 is positive or negative on each interval. If f 00 is positive on an
interval, then we know f is convex on that interval. If f 00 is negative, we know f is concave on that
interval.
If the function switches from convex to concave at ci , or if its switches from concave to convex
at ci , then ci is an inflection point. Otherwise, ci is not an inflection point.
0.2. Second derivative test. When we wished to find a local maximum or a minimum of f , we
began by solving f 0 (x) = 0. If c is a solution to this, we had a few methods of determining whether
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CALCULUS NOTES, WEEK PRECEDING SPRING BREAK
f (c) is a local maximum, a local minimum, or neither. If f is increasing to the left of c and decreasing
to the right of c, then we know f is biggest at c. So we cal fill out our increasing/decreasing table
and use this to determine where the maxima and minima occur. We could also plug in test points
on either side of c.
We next discuss the second derivative test, which (sometimes) provides us with another way of
answering this question. We assume f has two derivatives and that f 00 is continuous. Suppose that
f 0 (c) = 0 and that f is defined on both sides of c. We want to know if f (c) is a local maximum,
a local minimum, or neither. Suppose that f 00 (c) > 0. This means that f is convex in an interval
around f (since f 00 is continuous). But recall from the definition of convexity that this means the
tangent line to f at c must lie below the graph of f . But because f 0 (c) = 0, the tangent line is
horizontal (which means constant), and the constant value of the tangent line is f (c).
What does this mean? On an interval around c, the graph of f lies above the value f (c), which
means f (c) is a local minimum.
Similarly, if f 00 (c) < 0, f is concave around c, which means the tangent line L(x) = f (c) is above
the graph of f near c. This means f (c) is a local maximum. Thus we can summarize the second
deriative test as follows:
If f 0 (c) = 0, then
(i) if f 00 (c) > 0, f (c) is a local minimum,
(ii) if f 00 (c) < 0, f (c) is a local maximum,
(iii) if f 00 (c) = 0, the second derivative test is inconclusive.
We note that if f 00 (c) = 0, the test really is inconclusive. In this case, f (c) could be a local
maximum, a local minimum, or neither.
For example, if f (x) = x3 , f 0 (0) = f 00 (0) = 0, and f (0) is neither a local maximum nor a local
minimum.
If f (x) = x4 , f 0 (0) = f 00 (0) = 0, f (0) is a local minimum.
If f (x) = −x4 , f 0 (0) = f 00 (0) = 0, and f (0) is a local maximum.